Geometric Analysis and Topology Seminar

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A Bound on the Cohomology of Quasiregularly Elliptic Manifolds

Speaker: Eden Prywes, UCLA

Location: Warren Weaver Hall 517

Date: Wednesday, October 24, 2018, 11 a.m.

Synopsis:

A  classical result gives that if there exists a holomorphic mapping f : C ----> M, then M is homeomorphic to S^2 or S^1 xS^1, where M is a compact Riemann surface.  I will discuss a generalization of this problem to higher dimensions.   I will show that if M is an n-dimensional, closed, connected, orientable Riemannian manifold that admits a quasiregular mapping from R^n, then the dimension of the degree l de Rham cohomology of M is bounded above by the binomial coefficient n choose l.  This is a sharp upper bound that proves a conjecture by Bonk and Heinonen.  A corollary of this theorem answers an open problem posed by Gromov.  He asked whether there exists a simply connected manifold that does not admit a quasiregular map from R^n.  The result gives an affirmative answer to this question.